In Response To 'Re: Re: Re: Re: Strange NdSolve solution'
You might search your local library or abebooks.com for "Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations" by Forman Acton. That is a wonderful book on calculations and accuracy by one of the authorities in the field and I cannot recommend it too highly.
You might consider creating your own method of numerically solving differential equations. Some existing methods do an extremely good job near a particular point. Some do a good job at minimizing the relative error, error(f(x))/f(x) is minimized. But it sounds like what you are saying is e^100-error(e^100) is 12 and 12 is a huge number and I cannot accept an error of 12 for what I need to do. I believe the relative error is small and that is what most other people are concerned about.
And as I said previously, you might explore the world of accuracy and precision in Mathematica. This can be a very complicated subject and it can be difficult to track down exactly what you need to know to determine whether you can make NDSolve produce a result with an accuracy at e^100 that is acceptable to you. Sometimes this can be simple. Sometimes it is very difficult or perhaps nearly impossible to coax Mathematica to do what you need it to do. Sometimes it can take far far longer to do calculations with the accuracy needed to get the final result you require. And sometimes you may just have to write your own method to do this. But you may learn a lot in the process.